Worksheet 13: Conditional Probability#
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Poker. Each poker player gets \(5\) cards drawn from a deck without replacement. Let \(A\) be the event that your rival has at least one ace. Let \(B\) be the event that your \(5\)-card hand has all \(4\) aces in it.
a. What is the sample space? How many outcomes are there?
b. What is the probability of \(A\), \(\Pr[A]\)?
c. Suppose now we look at our cards and see that \(B\) happened. Given this new information, is it still possible that \(\Pr[A] > 0\)?
Take a moment to think of an example of an uncertain situation from real life, in which you learned information \(B\) that dramatically changed your estimate of whether some \(A\) was going to happen.
Why is the formula \(\Pr[A \mid B] = \frac{\Pr[A \cap B]}{\Pr[B]}\) a formula for the “zoomed in” probability of \(A\)?
Two coinflips. Suppose I flip a fair coin twice. Let \(A\) be the event that the first coinflip comes up heads, and let \(B\) be the event that at least one of the coinflips comes up heads.
a. What is \(\Pr[A \mid B]\)?
b. What is \(\Pr[B \mid A]\)?
Distracted Driving. Let \(A\) be the event that you are driving distracted, and let \(B\) be the event that you get in a car accident.
a. How would you phrase the statistic, “about 13% of car accidents involve distracted driving” in the language of conditional probability?
b. What is \(\Pr[B \mid A]\), in plain English?
c. What do you think happens more frequently: distracted driving, or car accidents?
d. Do you think \(\Pr[B \mid A]\) is smaller, larger, or no different than \(\Pr[A \mid B]\)?
The gateway drug. Consider the satement “Marijuana is a gateway drug: 9 out of 10 hard drug addicts tried marijuana first. \(B\) is the event of being a hard drug addict. \(A\) is the event of trying marijuana before trying hard drugs.
How would you phrase this in the language of conditional probability?
What is \(\Pr[B \mid A]\), in plain English?
Which do you think is more common: people trying marijuana, or hard drug addiction?
Do you think \(\Pr[B \mid A]\) is smaller, larger, or no different than \(\Pr[A \mid B]\)?
Testing for a disease. doctor orders a test for a patient to detect a rare disease. The test is 95% accurate. The disease affects 1% of the population. The test comes back positive.
a. How confident should the doctor be that the patient has the disease, given that the test came back positive?
b. Let \(A\) be the event that the patient has the rare disease. Let \(B\) be the event that the test is positive. The test is 95% accurate. How can we express the accuracy of the test in the language of conditional probabilities?
c. How can we express our confidence that the patient has the disease, given that the test is positive, in the language of conditional probabilities?
d. How would you express \(\Pr[\overline{A} \mid B]\) in plain English?